Thermodynamics of Associative Polymer Blends - Macromolecules

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

pubs.acs.org/Macromolecules

Thermodynamics of Associative Polymer Blends Debadutta Prusty,† Victor Pryamitsyn,† and Monica Olvera de la Cruz*,†,‡,§ Department of Materials Science and Engineering, ‡Department of Chemistry, and §Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, United States

Downloaded via UNIV OF NORTH DAKOTA on July 31, 2018 at 10:38:49 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

ABSTRACT: We report on a theoretical study of the phase behavior of polymer blends with hydrogen-bonding (HB) groups using the mean-field approximation. The hydrogen bonds are modeled as saturable bonds. We find the parametric conditions for lower and upper critical solution temperatures (LCST and UCST) and the eutectic behavior for the HB polymer blends. We quantify the conditions for the miscibility, partial miscibility, and immiscibility of HB polymer blends. We evaluate the conditions for the applicability of the Flory−Huggins theory to HB polymers by analyzing the composition dependence of the effective Flory−Huggins interaction parameter χeff. We use the polymer self-consistent field theory together with our model to compute the interfacial tension between coexisting phases in HB blends and find that intercomponent hydrogen bonding may significantly reduce the interfacial tension relative to a regular polymer blend at the same degree of segregation.

■

between hydrogen-bonding monomers.29,30 This model, also known as the attractive interaction model (AIM),31 has been used in SCFT studies to predict the phase behavior of various polymeric systems such as homopolymer−diblock copolymer blends and copolymer−copolymer blends. Part of the reason for its wide use in the literature is that it makes qualitatively correct predictions about the phase behavior in many systems, as corroborated by experiments.14,16 However, it is fundamentally incorrect to model hydrogen bonding by means of a negative χ parameter since χ parameter describes van der Waals type interactions, which are pairwise additive, nonsaturable, and spherically symmetric.32 On the other hand, hydrogen bonds are directional and saturable, implying that the formation of a hydrogen bond prevents the participating monomer from forming further bonds. Hence, there is a need to develop a model that treats hydrogen bonds as saturable. To this end, another class of models called association models have attempted to describe hydrogen bonding as a chemical reaction occurring between donor and acceptor groups. One of the earliest works on modeling hydrogen bonding in polymer blends using an association model is that by Painter et al.,33 who considered a blend of two homopolymers, each carrying one type of reacting unit. One of these units, namely B, contains both donor and acceptor groups and, hence, is capable of forming bonds with itself as well as with A. The other unit A can only associate with B. The self-association of B units, in a way similar to a linear polycondensation reaction, gives rise to linear chains (n-mers, Bn) of hydrogen-bonded units. The association of A unit with these chains leads to a product with the form BnA. The self-association and cross-

INTRODUCTION In the plastics industry, blending provides a simple and effective way to produce polymeric materials with improved mechanical and thermal properties over their individual constituent homopolymers. However, for most polymeric systems of practical interest, the efficacy of blending is thwarted by the little or zero entropy of mixing of long polymers and the unfavorable enthalpy of mixing coming from van der Waals interactions, leading to phase separation of the mixture. To overcome the above problem, research efforts in the previous two decades have extensively focused on improving the miscibility of blends through the introduction of favorable noncovalent interactions between the two components of the blend.1 In many systems, such interactions are affected by the presence of reacting functional groups/ stickers along the polymer chains. One example of such specific interaction is hydrogen bonding, which has been used extensively by researchers to alter the miscibility of blends.2−10 Hydrogen bonding has also been shown to produce a variety of ordered nanostructured morphologies in more complex systems such as diblock copolymer blends and solutions11−13 and homopolymer−copolymer blends.14−20 In addition, because of the reversible nature of hydrogen bonds, the extent of these interactions can be altered by changing temperature21 and pH.22,23 This phenomenon makes possible the fabrication of materials with highly tunable properties, such as selfhealing.24−28 In light of the above attractive features of polymeric systems with hydrogen bonding and a huge set of controlling parameters, there have been attempts to study the phase behavior of such systems using theoretical as well as molecular simulation methods. The simplest and most common approach to modeling hydrogen bonding involves using a negative Flory−Huggins parameter, χ, to describe the interaction © XXXX American Chemical Society

Received: March 29, 2018 Revised: July 11, 2018

A

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules association are characterized by equilibrium constants KB and KA, respectively. The free energy associated with the distribution of these hydrogen-bonded species, calculated using Flory entropy terms for these bonded chains, is the contribution of hydrogen bonding to the free energy. This approach is definitely an improvement over the physical models that use a negative χ parameter; however, it is cumbersome. Another limitation of this model is that it considers only those self-associating groups that have two hydrogen bond forming sites such as the carbonyl oxygen as acceptor and hydrogen attached to nitrogen as donor in urethane,34 which makes it possible to model association as linear condensation. In the case of groups with only one site, such as hydroxyl group in poly(4-vinylphenol),35 such an approach is not possible since once a self-associating hydrogen bond (B−B) has formed, the group cannot further bond with A. Therefore, we seek a simpler approach that can be generalized to all types of groups. To this end, we develop an association model to study the phase behavior of a blend of two homopolymers, namely A and B, containing hydrogen-bonding groups. Here, we consider a model that includes both self-association and interassociation of hydrogen-bonding groups. Our work is based on the meanfield theory of the solution of linear chains containing groups that associate in pairs developed by Semenov and Rubinstein.36 This approach was also used by Dormidontova37 to study the behavior of aqueous solutions of poly(ethylene oxide) (PEO) and water, taking into account both water−water and water− PEO hydrogen bonds. Since our approach is a mean-fieldbased approach, it cannot be used to address cooperativity in hydrogen bonding. Cooperative hydrogen bonding is generally found in crystalline or highly organized polymers such as cellulose,38 proteins,39,40 and DNA.41 However, since we consider an amorphous melt of polymers, the use of mean-field framework and the assumption of a decoupling of the polymer conformations and the sticker association remain valid. That is, in our model, hydrogen bonding does not change the Kuhn segments of the chains. In other words, the chains remain flexible (Gaussian). Besides studying the phase behavior, we also couple our model with self-consistent field theory (SCFT) to determine interfacial properties. Previously, SCFT has been used to study different hydrogen-bonding systems such as blends of homopolymers containing one or two hydrogenbonding groups at chain ends or along the backbone.42−45 In these studies, hydrogen bonding changes the chemical configurations, resulting in diblock, triblock, and graft copolymers from linear homopolymer chains. SCFT equations are then solved for each of these products to compute its density following the standard procedure. This approach is difficult to implement for polymers with multiple hydrogenbonding groups along their backbones since the number of new configurations grows factorially with the number of hydrogen-bonding groups. Along the lines of the above association model, Dehghan and Shi31 studied the phase behavior of a blend of AB diblock copolymer and C homopolymer with A and C interacting with each other via hydrogen bonding. The authors assumed that hydrogen bonding led to complete linear complexation of A and C blocks to give rise to a new diblock copolymer DB. Although their model is in agreement with experiments as regards the effect of C homopolymer fraction on the width of lamellas,15 it does not include all the degrees of freedom involved in bond formation. The entropy of bond formation coming from

various ways of forming hydrogen bonds between chains may contribute significantly to the free energy of the system. The model used in this work avoids the problem of determining the concentration of each species formed by associated stickers. In the association model, the free energy contains two separate contributions: the mixing free energy of unassociated chains and the interaction free energy due to hydrogen bonding, expressed in terms of the average fraction of functional groups in the associated form. An alternative approach, based on the Flory−Stockmayer model,46−48 is to calculate the partition function of clusters formed by associating stickers. Here, the mean-field free energy is obtained by minimizing the free energy with respect to the concentrations of these clusters. Semenov and Rubinstein36 derived, for a system of self-associating polymers in solutions, the mean-field free energy by means of the Flory−Stockmayer approach as well as an association model. The authors were able to show that both approaches led to the same expression for the free energy in the thermodynamic limit.36 The same conclusion was also reached by Bekiranov et al.,49 who studied hydrogen bonding between PEO and water molecules by means of a cluster model. Since we derive the free energy of our system based on the association model of Semenov and Rubinstein,36 we believe that our model correctly captures the thermodynamics. Another point to note is that the association model proposed by Semenov and Rubinstein36 is valid in both pre-thermoreversible-gel and post-thermoreversible-gel regimes and predicts that the reversible continuous gelation is not a thermodynamic transition. Unlike chemical cross-linking groups in polymers, thermoreversible bonds do not give rise to permanent gels since the system is still in the liquid phase and cannot support shear at zero frequency. Therefore, it is more apt to use the term “percolation” instead of gelation in the context of thermoreversible bonds. The percolation transition has previously been studied for self-associating polymers.36,50 Qualitative ideas about the percolation of clusters can be gained from the relationship between the number of functional groups on polymers and the fraction of reacted functional groups. According to the Flory−Stockmayer theory of gelation, for a solution of a polymer containing r functional units, the critical conversion ratio is given by pcrc = 1/(r − 1). For a melt of two homopolymers A and B,51 with rA and rB functional units that can form bonds of type A−B, the critical conversion 1 . ratios pAc and pBc are found to be proportional to (rA − 1)(rB − 1)

For our system with three kinds of bonds (two self-associating bonds and one interassociating bond), determination of critical conversion ratios can be cumbersome. Still these ratios (pAc, pBc, pABc) are expected to have an inverse relationship with the number of functional units per chain. Since in our system the number of functional units per polymer is high (for a polymer having a polymerization index 500 and a sticker fraction of 0.05, the number of stickers per chain turns out to be 25), these critical conversion ratios are small. Hence, we believe our system is mostly in a percolated state. Another aspect of phase behavior of systems containing associating units is the possibility of microphase separation. This has been predicted in binary melts of polymers with functional groups attached to the chain ends,42,43,51 where the products of association reaction can be diblock or triblock copolymers or networks having the lengths of each block equal to or greater than the corresponding unreacted block. In our system, since a chain contains several stickers, the average length of a strand of B

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules particular type (A or B) in the polymer network is likely to be less than the chain length. This reduces the possibility of microphase separation on a scale comparable to the chain length. Since we are mostly interested in the mixing to demixing transition, we do not pursue this subject in this paper. An advantage of our association model is that the effect of hydrogen bonding can be incorporated into the effective potential fields acting on monomers in SCFT without having to explicitly solve SCFT equations for each product. Our model can be used as a guide toward the design of polymers with composition-dependent miscibility.

bonding site. Hence, once a sticker participates in bond formation, it becomes unavailable for further bond formation. Our approach can also be adapted to systems where the groups have two hydrogen-bonding units such as urethane.34 Its derivation is given in Appendix B. Here, we consider the most general system where a sticker is capable of forming only one bond. All A* stickers belong to A polymer and all B* stickers to B polymer. The fraction of stickers in A and B chains are fA

■

and f B, respectively. Hence, the numbers of A* and B* stickers

MODEL AND THEORY Figure 1 shows a schematic of our system. We consider a mixture of A and B linear chains with degrees of polymer-

in the system turn out to be CA fAV and CB f BV, respectively. The energies of bonds are −ϵA, −ϵB, and −ϵAB for A*−A*, B*−B*, and A*−B* bonds, respectively. The negative signs are due to the attractive nature of interactions. The formation of hydrogen bonds results in a reduction of conformational entropy of the sticker monomers. Therefore, all stickers are not going to be in the associated form. Let the fractions of A* stickers participating in A*−A* and A*−B* bond formation be pA and pAB, respectively. The fraction of B* stickers participating in B*−B* bond formation is pB. There is no need to introduce an extra parameter for the fraction of B* stickers participating in B*−A* bond formation since the number of B* stickers involved in the formation of A*−B* bonds is the same as the number of A* stickers involved in the formation of A*−B* bonds. Now the number of pairs of A*−

Figure 1. Schematic representation of the system. Homopolymers A and B are represented by blue and red curves, respectively. The dots along the backbones represent the stickers. Circles are drawn around possible associative pairs.

A* stickers is NpA = pB NstB

pairs is NpB =

2

=

pA NstA 2

=

CBfB VpB 2

CAfA VpA 2

, the number of B*−B*

, and the number of A*−B* pairs

is NpAB = NAstpAB = CA fAVpAB. The factor of 2 in NpA and NpB ization NA and NB, respectively. We use the notation typically used in Flory−Huggins models for incompressible polymer melts.52 In this model, the length of the polymer segment 1 (Kuhn segment) is defined as b = 6ρ R 2 , where ρc is the c

comes from the fact that two stickers are needed to form a selfassociated pair. The free energy of the system is the sum of three independent parts: Fmix is the mixing entropy of unassociated

g

polymer chain number density and the radius of gyration of a

polymers, Fvan is the van der Waals interaction between nonsticker monomers, and Fst is the stickers’ contribution to

b2 N

polymer chain of N segments is R g 2 = 6 , where each segment has the mean-square end-to-end distance b2 and volume b3. We assume sterically symmetric chains bA = bB = b. The number of segments in every A chain is NA and in every B chain is NB. The enthalpic segment−segment interactions are characterized by the Flory interaction parameter between A and B segments χ > 0. The volume fractions of the A and B C C chains are ϕA = b3CA = CA and ϕB = b3CB = CB , respectively, 0

the free energy. Fmix and Fvan are from Flory−Huggins theory: fmix =

fvan =

0

where CA and CB are the number densities of polymer segments. C0 is the segment density of monomers and equal to 1 . The incompressibility ansatz can be presented as ϕA + ϕB = 3

where

b

1. Chains A and B contain randomly distributed associating groups/stickers of two types, A* and B*, respectively. Since we consider hydrogen bonding at the mean-field level without correlation, the distribution of these hydrogen-bonding groups along the chains does not affect the thermodynamics. We assume that each group/sticker has only one hydrogen-

ϕ ϕ ϕ Fmix ji ϕ zy = A logjjj A zzz + B log B j z VC0kBT NA NB NBe k NAe {

(1)

Fvan = χϕAϕB VC0kBT

(2)

Fvan/mix VC0kBT

give us the free energy per site. The stickers’

contribution to the free energy Fst is computed in the same manner as in the work of Semenov and Rubinstein.36 The detailed derivation is given in the Appendix. Here, we directly state the result C

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules ϕp f Fst = − A A A (log(ϕAλAfA ) − 1) VC0kBT 2 ϕBpB fB − (log(ϕBλBfB ) − 1) 2 ϕf − ϕApAB fA (log(ϕBλABfB ) − 1) + A A [pA log pA 2

In two-phase regions, the phase boundaries are found by equating the chemical potentials of each component in the two phases, i.e., μαA = μβA and μαB = μβB, where α and β are the two coexisting phases. It is worth noting that the free energy due to sticker association depends only on sticker fractions in the chains and has no dependence on the degree of polymerization of chains carrying these stickers. This actually allows us to use this model to study the phase behavior of other polymer architectures such as diblock copolymers. The only difference in the phase behavior in this case arises due to the translational entropy term, which changes due to the constraint of connectivity between two blocks. Interfacial Properties of the A|B Interface. In order to compute the interfacial tension and determine surface profiles, we define a quantity called excess free energy, which is the difference between the free energy of the system and that of the bulk phases and is computed as follows. For a free energy function, f(ϕA), we define the chemical potential as

fst =

+ 2pAB log pAB + 2(1 − pA − pAB ) log(1 − pA − pAB )] ÄÅ ij ϕBfB ÅÅÅÅ fA ϕA yzz j p zz + ÅÅÅpB log pB + 2jjjj1 − pB − ϕBfB AB zz 2 ÅÅ ÅÇ k { ÉÑ Ñ f ϕ ji zyÑÑ × logjjjj1 − pB − A A pAB zzzzÑÑÑÑ j ϕBfB zÑÑ k {ÑÖ ϵAϕAfA pA + ϵBϕBfB pB + 2ϵABϕAfA pAB − 2kBT (3)

where where λA = C0vA, λB = C0vB, and λAB = C0vAB are dimensionless constants and vA, vB, and vAB are the effective bond volumes of A*−A*, B*−B*, and A*−B* bonds, respectively. In the above equation, the first three terms subtract the ideal gas free energies of different types of bonded pairs, which have already been taken into account by eq 1. In principle, the association parameters ϵA, ϵB, ϵAB and λA, λB, λAB can be evaluated from the atomistic simulations of HB systems.53,54 The next two terms quantify the free energies of mixing of bonded and nonbonded stickers. The last three terms come from the energy gained during the bond formation. To obtain the equilibrium values of pA, pB, and pAB for a particular composition, we need to minimize the sticker free energy with respect to these three variables, which implies ∂fst ∂pA

∂fst

= 0,

= 0,

∂pB

∂fst ∂pAB

(

fA ϕA ϕBfB

2

pAB

)

fexc (ϕA) = f (ϕA) − f (ϕ0) − (ϕA − ϕ0)μA (ϕ0)

(

(1 − pA − pAB ) 1 − pB −

(5)

p

)

ij dϕA(x) yz 24ϕA(1 − ϕA)fexc [ϕA(x)] jj zz = jj zz b2 k dx { 2

(7)

From the above three conditions, we get a chemical equilibrium type condition relating pAB, pA, and pB.

(14)

The density profile ϕA(x) is determined by numerically solving eq 14 with the boundary conditions ϕA(x → −∞) = ϕ0 and ϕA(x → ∞) = ϕ1. Plugging eq 14 into eq 13 gives the expression for the interfacial tension

(8)

In order to study the phase behavior of blends, we write the free energy per site as f = fmix + fvan + fst

∫

The equilibrium concentration profile minimizes this interfacial tension. Hence, variational minimization of eq 13 leads to

= ϕBλABfB e ϵAB / kBT

ij f ϕ λAB2 yzji e 2ϵAB / kBT zy zzjj zzp p pAB2 = jjjj B B z j ϕAf λAλB zzjj e ϵA / kBT e ϵB / kBT zz A B { k A {k

2 | l o o ij dϕA(x) yz o b2 o o jj zz + f [ϕ (x)]o dx m } j z A o o exc j z o −∞ o o 24ϕA(1 − ϕA) k dx { o n ~ ∞

(13) (6)

ϕBfB AB

(12)

In order to compute the interfacial energy, we invoke the Helfand−Tagami approximation (HT), also known as ground state dominance approximation (GSD).52 Originally formulated by Helfand and Tagami56 to treat the case of two immiscible polymers in the limit χN → ∞ where fexc(ϕA) = χϕA(1 − ϕA), this method can also be used to determine the interfacial energy for an arbitrary functional form of excess free energy.55 Under this formulation, the interfacial energy per unit area of the interface can be written as

= ϕBλBfB e ϵB / kBT

fA ϕA

(11)

The excess free energy density is computed using the relation55

(4)

pAB

(10)

μA (ϕ0) = μA (ϕ1)

γ = b−3

pB

df dϕA

If ϕ0 and ϕ1 represent the compositions of the two coexisting phases, then the chemical potential satisfies the following relation:

=0

These three conditions give pA = ϕAλAfA e ϵA / kBT (1 − pA − pAB )2

1 − pB −

μA (ϕA) =

γ = 2b−3

(9)

∫ fexc [ϕA(x)] dx =

1 b

2

6

∫ϕ

ϕ1

0

fexc [ϕA ] ϕA(1 − ϕA)

dϕA (15)

D

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules The interfacial energy and the interfacial composition profile can also be determined by means of SCFT. The procedure to couple our model with SCFT is discussed in Appendix C.

■

RESULTS AND DISCUSSION Hydrogen bonds are directional and saturable, and we assume the persistence length does not change upon hydrogen bonding. Therefore, the system is locally homogeneous; that is, there is no local segregation of hydrogen bonds. This is in contrast to the case of random AB copolymers,57−59 where the interaction between A and B monomers is not saturable, leading to local segregation of A and B monomers depending on the strength of the interaction between the A and B monomers. In the context of Flory’s model, the phase separation tendency of a polymeric blend is governed by the χ parameter. For a mixture of two homopolymers, A and B, of equal degree of polymerization (N), if χN > 2, the mixture will spontaneously decompose into A-rich and B-rich phases; otherwise, it will remain homogeneous. In order to better understand the effect of hydrogen bonding on the phase behavior, it is useful to compute the effective χ parameter in our system, defined as χeff (ϕA) = χ −

Figure 2. Dependence of the effective Flory parameter, χeff(ϕA)N, on composition (ϕA) for different strengths of relative attraction (Δϵ). ϵ ϵ The plots are generated for χN = 20, k AT = k BT = 6.0, fA = f B = 0.05, B

association is stronger than interassociation (Δϵ > 0) and a decrease in the opposite case (Δϵ < 0). The former is a consequence of the increased tendency of polymer chains to be in the proximity of chains of the same type to form A*−A* and B*−B* bonds, which enhances their phase separation tendency. The latter is more interesting from the point of view of enhancing solubility. It can be seen in Figure 2 that the composition dependence of χeff(ϕA)N becomes stronger as Δϵ increases, and in a certain composition window, χeff(ϕA)N becomes less than 2 and even negative. In this particular symmetric case, this window is centered around the symmetric 1 composition (ϕA = 2 ) since the number of interassociation bonds is maximum when both stickers are present in equal amounts. It must be noted that such phase behavior cannot be modeled using a Flory model that uses a constant effective χ since that predicts miscible regions only in A-rich and B-rich regions. Previously, the concentration-dependent χ parameter has been obtained for polymer blends with dipolar interactions61 as well as for blends with strong electrostatic correlations.55,62 In the work of Kumar et al.,61 a mismatch in dielectric constants of two polymers caused the effective χ parameter to be dependent on composition and produced asymmetric phase diagrams even for a symmetric blend (NA = NB). Asymmetric phase diagrams were also theoretically obtained by Sing et al.62 for a blend of an uncharged homopolymer and a charged homopolymer when the electrostatic attraction strength between backbone charges and counterions became very strong. In such systems, a simple rescaling of the χ parameter is not possible, and phase behavior is different from that predicted by Flory−Huggins models. Not only does the phase separation tendency depend on the net energetic attraction/repulsion (Δϵ), but also it varies with relative fractions of stickers in A and B chains as well as bond volumes. For example, consider the case ϵ ϵ ϵ NA = NB = 500, k AT = k BT = 6.0, k ABT = 6.5. The fraction of

2 1 ∂ fst 2 ∂ϕA 2

= χ + χst

(16)

Here, χ is the Flory parameter describing the van der Waals interactions between monomers A and B, and χst is the change in χ caused by sticker interactions. If one were to model hydrogen bonding between A* and B* stickers using a negative Flory interaction parameter χAB, then χst would be equal to fA f BχAB. In our model, however, χst is not a constant, rather a function of composition. χeff(ϕA) can be determined from the structure factor of the blend, which is measured in scattering experiments on the homogeneous phase of the blend.60 The random phase approximation (RPA) for the miscible blend or a block copolymer in a disordered state52 relates χeff and the measured structure factor for a homogeneous composition ⟨ϕ⟩ through the relation 1 1 1 = A + B − 2χeff (⟨ϕA⟩) S(q) So (q) So (q)

(17)

Although the effective χ parameter cannot directly be used to determine the binodals, it allows one to find the spinodals from eq 17. Therefore, it is instructive to study the composition dependence of χeff(ϕA) for different parameters in our model. First, we consider a symmetric blend with NA = NB = 500 and fA = f B = 0.05 and vary ϵAB , keeping both ϵA and ϵB constant. kBT

kBT

kBT

We characterize the net attraction or repulsion between A and 1 ϵ +ϵ B by a parameter Δϵ = k T A 2 B − ϵAB . Plots corresponding B

(

B

and NA = NB = 500. When Δϵ is positive, self-association is stronger than interassociation, and when Δϵ is negative, interassociation is stronger than self-association.

B

B

B

stickers in A is kept constant at fA = 0.1. Figure 3 plots χst as a function of composition for different values of f B. When f B = 1 fA, χst attains a minimum at ϕA = 2 , which was also observed in Figure 2. However, this symmetry breaks down when f B ≠ fA. When f B < fA, the minimum in χst is located in the A-lean region, and χst increases with ϕA. Since B* stickers are in the minority here, for sufficient concentration of A, A* stickers are more likely to bond with themselves even though A*−B* bonds are energetically more favorable. This leads to an

)

to different values of Δϵ are shown in Figure 2. When ϵA, ϵB, and ϵAB are equal, χeff(ϕA) is constant across the entire range of composition. In this case, the energies of self-association are equal to that of interchain association, resulting in zero effective attraction. Hence, χeff(ϕA)N is equal to the intrinsic χN and independent of concentration. As we increase the net attraction or repulsion, χeff(ϕA)N starts exhibiting a composition dependence. There is an increase in χeff(ϕA)N when self-

E

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

bonds are energetically equally likely to form, a mismatch in λ values can lead to an enhancement/suppression in phase separation tendency. When λB < 0.04, the entropic cost of forming self-associated bonds becomes higher, causing χst to be negative. The reverse behavior is observed when λB > 0.04, indicative of enhanced tendency of segregation. The dependence of χst on composition can lead to significant modifications to the phase diagrams of polymer blends. For a neutral binary blend, the χN vs ϕA phase diagram is symmetric around ϕA = 0.5, and the critical point corresponds to χN = 2. We first analyze the effect of interattraction strength on the phase behavior in a symmetric system (NA = NB = 500, λA = λB = λAB = 0.04, and fA = f B = 0.05). The results are shown in Figure 5. When Δϵ > 0, the phase diagram shifts vertically

Figure 3. Correction to the Flory parameter, χstN, as defined in eq 16, as a function of composition (ϕA) for different degrees of asymmetry in sticker fractions. The fraction of stickers in A, fA, is held constant at 0.1, and the fraction of stickers in B, f B, is varied from 0.05 to 0.2. The ϵ ϵ other parameters are fixed at Δϵ = −0.5, k AT = k BT = 6.0 , and NA = B

B

NB = 500.

increase in χst with ϕA. The opposite trend is observed for f B > fA. Figure 4 shows the dependence of the correction to the effective Flory parameter arising from hydrogen bonding, χst,

Figure 5. χN vs ϕA phase diagram for various strengths of relative ϵ ϵ attraction (Δϵ) at kTA = kTB = 6.0, fA = f B = 0.05, and NA = NB = 500. When self-association and interassociation are equally probable (i.e, Δϵ = 0), the obtained phase diagram is the same as the phase diagram for a neutral symmetric blend. For cases where self-association dominates cross-association (Δϵ > 0) or cross-association is dominant but weak in magnitude, the phase diagram shifts vertically down or up as depicted in (a). However, in the limit of stronger cross-association, two two-phase coexistence regions are seen with a miscible region in the middle of the composition axis as shown in (b). To explain this behavior, we plot, in Figure 7, the effective Flory interaction parameter (χeff(ϕA)) vs composition(ϕA) for χN values in (i) one two-phase coexistence region (shown by the horizontal bar in (a) for Δϵ = −0.5) and (ii) two two-phase coexistence regions (shown by the horizontal bar in (b) for Δϵ = 1.5).

Figure 4. Effect of asymmetry in effective bond volumes on χst(ϕA)N. ϵ ϵAB ϵ = 6.0, fA = f B = 0.05, and NA Plots were obtained for kTA = kTB = kT = NB = 500. λA and λAB are kept constant at 0.04. If λB < λAB, the entropic loss accompanying the formation of B*−B* bonds is higher than that associated with A*−B* bonds. The converse is true for λB > λAB.

on the asymmetry in bond volumes. We consider the case ϵ ϵ ϵ where kTA = kTB = kTAB = 6.0, which represents zero net attraction. Since the dimensionless constant λi (i = A, B, and 1 AB) is the product of monomer concentration (C0 = 3 ) and b the effective bond volume (vi), a higher value of λi signifies a higher probability of bond formation. These effective bond volumes take into account the probability that the orientations of the donor and the acceptor are in the proper alignment. Dormidontova and ten Brinke63 obtained an expression relating the effective volume to the physical volume, which is v 1 − cos Δi given by vi = . Here, Δi is the maximum allowed 2 deviation from the orientation in which the two groups point toward each other. This expression was originally derived for water−water and water−PEO hydrogen bonds and captures the essential physics involving directional bonds in general. The characteristic angles can be determined experimentally π π and lie in the range 8 − 5 for the water−PEO system.37 For

down, indicating an increased degree of phase segregation. At sufficiently high values of Δϵ, the phase separation can take place even in the absence of short-range van der Waals interactions (χN = 0). In the situation where the interattraction dominates over self-association (Δϵ < 0), the phase diagram moves upward, indicating improved mixing between two homopolymers. For the case of stronger selfassociation (Δϵ > 0) as well as for the case where the interassociation is slightly greater than self-association, the shift is vertical and the shape of the phase diagram remains unchanged from that of the neutral system. Hence, the phase behavior can be described by a constant effective χ parameter. However, when the interassociation becomes sufficiently strong (Figure 5b), in addition to shifting vertically upward, the phase diagram exhibits two two-phase coexistence regions with a miscible region in between. We notice the same trend when we increase the fraction of stickers while keeping the interaction strength constant at Δϵ = −0.5 (Figure 6). The observed behavior can be explained by looking at the behavior of concentration dependent χeff(ϕA) . Figure 7 shows the χeff(ϕA)N vs ϕA plots for the labeled values of intrinsic χ in the weak (Δϵ = −0.5) as well as strong (Δϵ = −1.5) interassociation bonding limits, respectively. When the hydro-

our calculations, we use a characteristic angle of π , which leads 8 to vhb ≈ 0.04b3 and λ = 0.04, where we assume v = b3. We fix λA = λAB = 0.04 and vary λB. Please note that we are not focusing on any particular system, and these values are for calculation purposes only. From the figure, it is clear that even though all

F

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 8. Effect of sticker fraction asymmetry on χN vs ϕA phase diagrams. The other parameters are fixed at ϵ ϵ ϵA = k BT = 6.0, k ABT = 7.0 , NA = NB = 500, and fA = 0.1. For the k T

Figure 6. Effect of sticker fraction on χN vs ϕA phase diagram. The ϵ ϵ plots were obtained for kTA = kTB = 6.0 , Δϵ = −0.5, and NA = NB = 500. For low sticker fractions, the phase diagram looks similar in shape to that for neutral blends with its critical point shifted vertically up. However, when the amount of stickers in chains is significant (blue and purple curves), we find two two-phase coexistence regions in the phase diagram. At higher sticker fractions, the effects of crossassociation become strongly composition dependent, leading phase diagrams that cannot be described by a constant effective interaction parameter in a Flory-type model.

B

B

B

symmetric case (f B = fA = 0.1), we see enhanced miscibility in the middle portion of the composition axis. However, this does not remain the case for asymmetric systems. When f B < fA, we observe that the region of maximum miscibility is located in the A-lean (low ϕA) region and the miscibility is decreased in the A-rich region (as evident in the downward shifting of the phase diagram in this region). An exactly opposite trend is observed when f B > fA. 1

maximum for the symmetric composition (i.e., ϕA = 2 ), which is due to the maximum number of A*−B* bonds at this composition. This is corroborated by the minimum in χst at 1 ϕA = 2 for the symmetric case in Figure 3. Bringing asymmetry into the system makes the phase diagram asymmetric. This shift in critical point comes down to the imbalance in stoichiometry of A* and B* stickers. When f B > fA, at low ϕA, there is a surplus of B* stickers, which leads to self-association of B* stickers that are not bonded with A* stickers. This explains the increased phase segregation tendency in this region. On the other hand, when ϕA is high, there are sufficient A* stickers to bond with B*, and χst attends a minimum in this region as has been shown by black and purple curves in Figure 3. This results in the improved miscibility in this region. The same argument can be applied to the case where fA > f B. A Flory parameter, χN, vs composition diagram provides useful information about phase behavior of different polymeric systems (with different χ values) with the same hydrogenbonded groups. However, in order to study the thermoresponsive behavior of polymers, it is useful to examine their temperature vs composition phase diagrams. Here, we consider a symmetric blend of polymers with NA = NB = N = 500, λA = λB = λAB = 0.04, and fA = f B = 0.05. We vary the interassociation energy ϵAB, keeping the self-association energies constant at ϵA = ϵB = 1800k = 6kBTr, where Tr (300 K) is the room temperature. As for the dependence of χ on T, we use experimental values for PS−PMMA 1.96 χ = 0.0129 + T .64 Although the above expression was obtained experimentally over the temperature range 100− 200 °C, we use it to construct phase diagrams over the range 200−500 K since we are not focusing on any specific system but are interested in the general behavior of the system. The results are shown in Figure 9. The above combination of χ and N represents a highly segregated system. In the absence of any cross-association, there is little miscibility over the entire temperature range considered here. On increasing Δϵ, we observe that the blend becomes miscible at low temperatures 1 near the symmetric composition (ϕA = 2 ) as evident by the formation of a miscibility loop (shown in red). This loop widens and extends toward higher temperatures as we further

Figure 7. Variation of the effective interaction parameter, χeff(ϕA)N with composition for two cases. The first case pertains to Δϵ = −0.5 and χN = 7.5. This combination falls in the two phase region in χN vs ϕA diagram and is represented by the horizontal bar in Figure 5a. In this case, χeff(ϕA)N is greater than 2 across the entire composition range as depicted by the blue curve. Hence, we obtain a phase diagram with a miscibility gap in the middle portion of the composition axis. This is similar in shape to the phase diagram predicted by a Flory-type model. In the second case, Δϵ = −1.5 and χN = 21.7. Here, we find two two-phase coexistence regions (as shown by the horizontal bar in Figure 5b). Here, χeff(ϕA)N shows a strong composition dependence and becomes less than 2 in the middle portion of the composition axis. This explains enhanced miscibility in this region and immiscibility in both A-rich and B-rich regions.

gen bonding is weak, the variation in χ eff(ϕ A ) with concentration is weak. Hence, χeff(ϕA)N always remains above 2 at the critical composition in the two-phase region. In the limit of strong interassociation strength (Δϵ = −1.5 in Figure 7) or high sticker fraction, χeff(ϕA)N acquires a strong concentration dependence and becomes less than 2 and even negative in the middle of the composition axis. This miscibility range becomes narrower as χN increases since the van der Waals repulsion between monomers increasingly counteracts interattraction energy. Finally at a certain value of χN (around 25 in Figure 5b, green curve), this region vanishes, and the two two-phase regions merge into a single two-phase region in an eutectic-like manner. Previously, eutectic-like spinodal curves were predicted by Mester et al.51 for a blend of star polymers, A and B, with associating functional groups at the terminus of each arm for sufficient bonding strengths. Figure 8 shows the effect of sticker asymmetry on χN vs ϕA phase diagrams. In the symmetric system, the solubility is G

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 9. Temperature vs composition phase diagrams for various relative strengths of attraction. These plots are for large values of molecular weight (NA = NB = 500). The other parameters are ϵA = ϵB 1.96 = 6kBTr, χ = 0.0129 + T , fA = f B = 0.05, and λA = λB = λAB = 0.04, where Tr is the room temperature (300 K). The dashed lines are tie lines in regions of phase coexistence. This system represents a highly segregated blend. For small values of ϵAB, we see miscible loops in the middle with two-phase regions in A-rich and B-rich regions, which is shown in (a). As the strength of interassociation increases, this loop widens, finally opening a whole region of miscibility across the entire composition, as shown in (b).

Figure 10. Temperature vs composition diagrams for various relative strengths of attraction. These plots are for short polymers (NA = NB = 1.96 120). The other parameters are ϵA = ϵB = 6kBTr, χ = 0.0129 + T , fA = f B = 0.01, and λA = λB = λAB = 0.04. Here, we notice closed immiscibility loops. At low temperature, hydrogen bonding is quite strong, which makes the blend miscible. Above LCST, hydrogen bonding is not strong enough to counteract van der Waals repulsion, leading to immiscibility. However, since the polymer is short, we also see an UCST since van der Waals repulsion weakens as temperature increases.

parameter from it using the relationship between w and χ developed by Helfand and Tagami.69 Here, we show that the interfacial properties cannot be accurately predicted by a Florytype model with a constant effective χ. In other words, two different models can make different predictions about interfacial properties between the same coexisting phases. To this end, we give a comparison between interfacial properties predicted by our association model (AM) and a Flory interaction model (FM) with an effective χ. In other words, we first compute the coexisting phase compositions from AM and then find an effective χ in the equation

increase ϵAB as shown by the green curve. At a very high value of ϵAB (shown by the purple curve), this loop vanishes and the blend becomes miscible across the entire composition range for intermediate temperatures. At high temperatures, the blend becomes immiscible. Enhanced miscibility at low temperatures and immiscibility at high temperatures is common in systems with specific interactions. In many systems, it is manifested by a lower critical solution temperature (LCST) such as in PEO− water65 as well as several miscible polymer blends such as poly(hydroxy ether of bisphenol A) and poly(ϵ-caprolactone)66 as well as poly(methyl methacrylate)−poly(styrene-coacrylonitrile).67 At low temperatures, the van der Waals repulsion is overwhelmed by the favorable interaction that exists between two components due to interassociation, making the blend miscible. As the temperature increases, the effect of interassociation becomes weaker as can be seen in the free energy expression (eq 3). Hence, the blend becomes immiscible at high temperatures. Since it is the balance between the driving force for mixing arising from crossassociation and entropy of polymer chains and unfavorable van der Waals repulsion that governs the phase behavior, different combinations of N and fA and f B will lead to different phase diagrams. Shown in Figure 10 are the phase diagrams for shorter copolymers (N = 120) with fA = f B = 0.01. Here, one finds both LCST and upper critical solution temperature (UCST) in phase diagrams, giving rise to an immiscibility loop. These features are reminiscent of the phase coexistence loops in aqueous solutions of polymers forming hydrogen bonds with water. Closed phase coexistence loops for PEO in water were explained by Dormidontova37 by means of an association model and by Matsuyama and Tanaka68 and Bekiranov, Bruinsma, and Pincus49 within the framework of a cluster model. The interfacial energy of the interface separating two coexisting phases in polymeric systems influences the morphology of microphases as well as the size of the minority phase in the majority matrix. Often interfacial energy and hence the interfacial width between two homopolymers are linked to the effective χ parameter. In fact, one of the experimental ways of determining χ parameter between two homopolymers involves measuring the interfacial width (w) by means of neutron reflectometry and computing the χ

fFlory =

ϕA NA

ϕA

ϕB

NA

NB

( )+

log

ϕB

( )+χ

log

ϕ ϕ that produces

Flory A B

NB

the same coexisting phase compositions. For a symmetric blend with a homopolymer degree of polymerization N, where ϕA and 1 − ϕA are the compositions of the coexisting phases, χFlory is calculated by χFlory =

log(ϕA /(1 − ϕA)) (2ϕA − 1)N

(18)

Since the FM does not predict two two-phase regions in the phase diagram, we restrict our analysis to the region in the phase diagram where there is only one two-phase region and the phases in equilibrium are symmetric about ϕA = 0.5. We plot the interfacial energy as a function of χN for various strengths of interassociation (Δϵ) in Figure 11. This interfacial energy is normalized by γ0, which is the interfacial tension of a neutral blend of infinite N corresponding to the same coexisting composition. γ0 is computed using the relation γ0 = bρ0 kBT

χFlory 1/2

( ) 6

. The ratio

γ γ0

becomes 1 in the limit

χFloryN = ∞, which happens when χN → ∞. In this limit χFlory ≈ χ because the contribution from hydrogen bonding to the free energy is negligible compared to the contribution from repulsive interactions due to χ. The plots are generated using both SCFT and GSD (eq 15). It is seen that both GSD and SCFT produce the same trend with regard to the variation of γ γ0

with χN although GSD gives slightly higher values. GSD is accurate for a very high degree of segregation. However, because it is fast and computationally straightforward compared to an SCFT simulation, it can be used to gain H

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

Figure 11. Normalized interfacial energy ( γ ) between A-rich and B-rich phases vs χN for various degrees of interassociation strengths. γ has been γ0

computed using both ground-state dominance (GSD) (eq 15) and SCFT calculations. Here, γ0 refers to the interfacial energy of a neutral blend with the same coexisting composition, computed analytically using the Helfand−Tagami formula γ0 = bρ0 kBT

χeff 1/2

(6)

, where χeff has been defined

in eq 18. AM refers to association model, which has been developed in this paper. FM refers to a Flory-type model that produces the same ϵ ϵ coexisting composition as AM. These plots are generated using k AT = k BT = 6.0 , fA = f B = 0.05, and NA = NB = 500. The interassociation strengths B

B

are (a) Δϵ = −0.2, (b) Δϵ = −0.5, (c) Δϵ = −1.0, and (d) Δϵ = −1.5. The surface profiles corresponding to the vertical lines are shown in Figures 12 and 13.

qualitative ideas about interfacial properties in all segregation regimes. The other advantage of GSD is that it helps us correlate interfacial properties with the nature of excess free energy. Returning to the discussion of Figure 11 results, we see that when Δϵ is low (i.e., the hydrogen-bonding strength is weak), the interfacial energy values predicted by the two models (AM and FM) are close to each other (Figure 11a,b). However, as the bonding becomes stronger, we observe that the difference between these values becomes prominent (Figure 11c,d). Here, the AM predicts a lower value of γ

Figure 12. Composition profiles across the A−B interface generated by using ground-state dominance for two cases (a) Δϵ = −0.2 and χN = 4.8 and (b) Δϵ = −1.5 and χN = 24.8. The fixed parameters are ϵ ϵA = k BT = 6.0 . In the former interassociation strength is weak, k T

γ0

than the FM. Figure 12 shows the composition profiles of A across the interface (generated using GSD) using both models in the weak as well as strong hydrogen-bonding limit. As observed in the case of the interfacial tension, the composition profiles merge when the hydrogen bonding is weak. When the hydrogen bonding is strong, our model predicts a more diffuse interface. In addition, this interface also differs in shape from that predicted by the FM with the presence of more inflection points in the former. The composition profile in the FM can be described by the equation ϕA(x) = ϕ0 +

| ϕ1 − ϕ0 l o jij 2(x − x0) zyzo o o 1 + tanh m } j z j z o o o 2 n w k {o ~

B

B

while in the latter there is significant interassociation.

where x0 is the location of the interface, w is the width of the interface, and ϕ0 and ϕ1 are the compositions of the coexisting phases. However, the above equation fails to fit the profile predicted by the AM. The above predictions are also corroborated by SCFT numerical simulations as shown in Figure 13. In order to explain the difference in predictions about interfacial properties by the above two models, we plot in Figure 14 the excess free energy vs composition curves for both models corresponding to the above surface profiles. It is seen that when the strength of interassociation is low, both free

(19) I

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

observed phase behavior is in contrast with the predictions made by models that make use of a constant χeff, which is the sum of the bare Flory−Huggins interaction parameter and a constant negative interaction parameter (χst) describing the interaction between functional groups. For a blend where the degrees of polymerization of the two components are equal, the interaction models predict either a fully miscible blend if χeffN < 2 or an immiscible blend with the coexisting 1 compositions located symmetrically with respect to ϕA = 2 if χeffN > 2. On the other hand, in the case of a symmetric blend of polymers of equal degree of polymerization and equal sticker fractions, the association model predicts enhanced 1 miscibility around the symmetric composition (ϕA = 2 ) where χeff is minimum due to the maximum number of interassociation bonds forming at the symmetric composition. Making the sticker fractions in polymers unequal makes the phase diagram asymmetric and results in enhanced miscibility in Arich or B-rich regions. This asymmetry is attributed to the imbalance in the number of stickers in A and B chains, which shifts the composition of maximum cross-association from 1 ϕA = 2 . We have also constructed temperature vs composition diagrams for such systems. We notice enhanced miscibility at low temperatures as the effects of hydrogen bonding overwhelm the van der Waals repulsion at lower temperatures and become weak as the temperature increases. With the caveat that our approach neglects fluctuations, which are small in blends near critical points only when the chain molecular weight is sufficiently large,70−72 for blends of short polymers, we also find the existence of both LCST and UCST, leading to an immiscibility loop in the phase diagram; note that contrary to microphase-segregated copolymers, where fluctuations destroy the critical points,73,74 blends of chains with finite molecular weights do have critical points.70,72 Our study reveals that introducing functional groups along the polymer backbone offers considerable latitude in tuning the phase behavior and miscibility of polymer blends. The difference between the AM and the FM becomes further evident when we determine the interfacial tension and the composition profile across the interface between two immiscible polymers by using the ground-state dominance approximation. For the same coexisting compositions, taking into account the saturable nature of hydrogen bonding by the use of an association model led to a lower value of surface tension and a broader interface than those predicted by the FM. This difference is attributed to the modification in the shape as well as the magnitude of free energy vs composition curve that results from an elaborate treatment of sticker association. Our work shows that even for highly segregated blends, we can lower the interfacial tension by introducing favorable interactions among polymers. This result has implications in the blending process since the droplet size of the minority phase in polymer blends is directly proportional to the interfacial tension.75 We have also incorporated our model into SCFT of blends. The composition profile determined using SCFT support the results of ground-state dominance. Since our model produces a composition-dependent χ, it will be interesting to use this model to investigate the impact of hydrogen bonding on morphologies, coming from the saturable nature of bonds, in more complex systems such as diblock copolymer blends and diblock copolymer−homopolymer blends.

Figure 13. Composition profiles across the A−B interface generated by SCFT simulations. The parameters for these plots are the same as those for Figure 12. Our SCFT results support the results obtained using ground-state dominance.

Figure 14. Excess free energy vs composition curves corresponding to the AM and the FM for profiles in Figure 11. In (a), Δϵ = −0.2. Here, the interassociation strength is weak. Hence, the free energy curves are similar in shape and magnitude. In (b) Δϵ = −1.5. A significant amount of interassociation in this case leads to a big difference in the shapes of free energy curves produced by two models. Plots (a) and (b) provide an explanation for the difference in interfacial properties predicted by two models.

energy curves have the same shape. However, in the limit of strong interassociation, the two free energy curves differ considerably. In this case, the excess free energy plot of the AM displays a well centered around the middle of the composition axis. This decrease in the free energy is due to the maximum number of interassociation bonds forming when the fractions of both components are equal. Modeling the phase behavior using a constant χFlory produces a convex upward curve and misses out on these subtle changes in the thermodynamics. Since the interfacial energy depends not only on the composition of the existing phases but also on the magnitude of the excess free energy, we see a noticeable difference in the interfacial tension and composition profiles even for the same composition of coexisting phases.

■

CONCLUSIONS We have studied the phase behavior of blends of polymers with two types of functional groups that are capable of forming hydrogen bonds with groups of the same type as well as with groups of the different type. We use an association model that treats the hydrogen bonds as saturable bonds to derive the free energy of the system in the mean field limit. We construct phase diagrams of the system for various combinations of parameters such as fractions of functional groups in polymers, the strength of hydrogen bonding, and the asymmetry in sticker fractions. We invoke an effective interaction parameter, χ eff , to qualitatively explain the phase behavior. We demonstrate that in the limit of significant attraction between the two functional groups the effective interaction parameter, χeff, acquires a strong composition dependence. This leads to composition-dependent solubility and multiple two-phase coexistence regions in the χN vs ϕA phase diagram. The J

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

■

APPENDIX

the entire product by NpA! to avoid overcounting. Thus, the number of ways of selecting 2NpA stickers is

A. Free Energy of the Stickers Association

In this section, we derive an expression to quantify the contribution of hydrogen bonding to the overall free energy. The partition function of the system, following the approach of Semenov and Rubinstein,36 can be written as Z = Zref Zst e−χϕAϕBC0V

PA − AA =

(20)

ij ϕ yz ij ϕ yz ϕ ϕ log(Zref ) = A logjjj A zzz + B logjjj B zzz j z j NBe z VC0kBT NA NB k NAe { { k

PA − AB =

(22)

(23)

Pcomb =

Since our system is a melt, it is safe to neglect any correlations between sticker positions. Therefore, the probability that all these pairs are close enough as well as in the correct orientation to form bonds is given by iv y W = jjj A zzz kV {

NpA

ij vB yz jj zz kV {

NpB

ij vAB yz jj zz kV {

×

NpAB

NpAB! (NstA − 2NpA − NpAB)!

(27)

NstA! NpA ! NpAB! (NstA − 2NpA − NpAB) ! 2 NpA NstB!

NpB! (NstB − 2NpB − NpAB) ! 2 NpB

(28)

The free energy density is expressed as

log Pcomb log W Fst 1 ijj ϵACApA fA =− − − j VkBT V V kBT jjk 2

(24)

where vA, vB, and vAB are the effective bond volumes of A*−A*, B*−B*, and A*−B* bonds, respectively. Now, the combinatorial factor Pcomb is the number of possible microstates consistent with the macrostate corresponding to NpA A*−A* bonds, NpB B*−B* bonds, and NpAB A*−B* bonds. This is expressed as Pcomb = PA − AAPA − ABPB − BBPB − ABPpair − AB

(NstA − 2NpA)!

The corresponding factors PB−BB and PB−AB can be calculated similarly. Now, the last term Ppair−AB, which accounts for the number of ways of forming AB bonds between NpAB A* stickers and NpAB B* stickers is given by NpAB!. Therefore, the combinatorial factor becomes

Zst is the partition function describing the sticker interactions and given by Zst = PcombW e(ϵANpA +ϵBNpB+ϵABNpAB)/ kBT

2 NpANpA ! (NstA − 2NpA)!

Now, the number of ways of selecting NpAB stickers out of the remaining (NAst − 2NpA) A* stickers is

The contribution from van der Waals repulsion is given by −kBT log(e−χϕAϕBC0V ) = χϕAϕB VC0kBT

NstA! (26)

(21)

fvan =

2 NpANpA! =

where Zref is the partition function of the system of noninteracting polymers. It is responsible for the entropic contribution to the free energy, given by fmix = −

NstA(NstA − 1)(NstA − 2)...(NstA − 2NpA + 1)

+

ϵBCBpB fB 2

yz + ϵABCApAB fA zzz z {

(29)

Expressing NAst, NBst, NpA, NpB, and NpAB in terms of CA and CB and using the Stirling’s approximation log N! ≃ N log N − N, we obtain

(25)

ϕp f Fst ji ϕ λAf zy = − A A A logjjj A A zzz j e z 2 VC0kBT k { ϕBpB fB ji ϕ λBf zy ji ϕ λABf zy − × logjjj B B zzz − ϕApAB fA logjjj B B zzz j z j e z 2 k e { k { ϕAfA + [p log pA + 2pAB log pAB + 2(1 − pA − pAB ) 2 A ÄÅ ϕBfB ÅÅÅÅ ÅÅp log p × log(1 − pA − pAB )] + B 2 ÅÅÅ B ÅÇ ÑÉ ij fA ϕA yzz ijj fA ϕA yzzÑÑÑÑ j p zzlogjj1 − pB − p zzÑÑ + 2jjj1 − pB − j ϕBfB AB zz jj ϕBfB AB zzÑÑÑ k { k {ÑÖ

Here, PA−AA is the number of ways of choosing NpA pairs out of NAst A* stickers for A−A bonds, PA−AB the number of ways of choosing NpAB A* stickers out of the remaining NAst − 2NpA A* stickers for A−B bonds, PB−BB the number of ways of choosing NpB pairs out of NBst B* stickers for B−B bonds, PB−AB the number of ways of choosing NpAB stickers out of the remaining NBst − 2NpB B* stickers for A−B bonds, and Ppair−AB the number of ways of pairing NpAB A* stickers with NpAB B* stickers for A−B bonds. We compute each term below. In order to compute the first term, we note that the first sticker can be chosen in NAst number of ways. The second sticker, which the first sticker will associate with, can be selected in NAst − 1 ways. However, since the order of selection of these two stickers does not matter, the number of ways this first pair can be chosen is NAst(NAst − 1)/2. Similarly, out of the remaining Nst − 1 stickers, the second pair can be chosen in (NAst − 2)(NAst − 3)/2 ways. Continuing like this, we find that the NpA the pair can be chosen in (Nst − 2(NpA − 1))(Nst − 2(NpA − 1) − 1)/2 number of ways. We again note that the order of these NpA pairs is inconsequential; that is, selecting a pair, let us say pair 1 first and a pair 2 second is the same as selecting pair 2 first and pair 1 later. Hence, we need to divide

fst =

−

ϵAϕAfA 2kBT

pA −

ϵBϕBfB 2kBT

pB −

ϵABϕAfA pAB kBT

(30)

where λA = C0vA, λB = C0vB, and λAB = C0vAB are dimensionless constants. K

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

ji λBf ϕ zy ji λABfB zyz fst = −ϕBfB pB logjjj B B zzz − ϕBfB pBA logjjj j e zzz j e z k { k {

B. An Extension for the Groups Containing Two Hydrogen-Bonding Sites

Here, we describe how our model can be extended to units containing two hydrogen-bonding sites used in the Painter model.33 One of the most common cases encountered in experimental literature is a blend where one polymer contains groups that have a donor group and an acceptor group and hence can self-associate such as urethane and poly(vinylphenol). The other component has a group that cannot self-associate but can only form bonds with one of the groups of the first polymer. An example is the nitrogen group of vinylpyridine.10 Let us call the unit capable of both selfassociation and interassociation B*. The total number of B* stickers becomes NBst = CBVf B. Since each B* contains one donor and one acceptor, the numbers of B*-type donors and acceptors in the system are also equal to CBVf B. The group capable only of interassociation is denoted by A*. The number of such stickers is NAst = CAVfA. We define two quantities pB and pBA as the fractions of B* stickers forming self-associated bonds and interassociated bonds, respectively. Hence, the numbers of self-associated pairs and interassociated pairs are NBB = CBVf BpB and NAB = CBVf BpBA, respectively. The procedure for deriving the sticker free energy remains the same as in the previous section. The only difference in this case is the calculation of the combinatorial factor, Pcomb, which can be expressed as Pcomb = PB1 − BBPB2 − BBPBpair − BBPB1 − ABPA − ABPpair − AB

+ ϕBfB [pB log pB + (1 − pB ) log(1 − pB ) ij yz ij yz ϕf ϕf ϵ + ϕAfA jjjj1 − B B pBA zzzz logjjjj1 − B B pBA zzzz − B ϕBfB pB j z j z ϕAfA ϕAfA k { k { kBT ϵBA − ϕf p kBT B B BA (38) + (1 − pB − pBA ) log(1 − pB − pBA )]

C. SCFT for Associative Polymers

In this section, we discuss the method of incorporating our theory into SCFT. Although, we do not need SCFT to analyze the phase behavior of blends, it can provide a useful verification of the results of ground-state dominance. In addition, for exploring more complex morphologies, such a simple free energy analysis is not possible. In such cases, one has to rely on SCFT to investigate the phase behavior and interfacial properties. To perform SCFT, we recall that the free energy per chain in a blend of A and B homopolymers having a degree of polymerization N in the canonical ensemble formalism is a functional of density profiles and external fields acting on individual segments and is given by76 ÄÅ ÉÑ ÅÅ i Q y ÑÑ jj A zz NF F Ñ 0Å Å = = −ϕA ÅÅÅlogjjj 0 zzz + 1ÑÑÑÑ ÅÅ j ϕA V z ÑÑ C0kBTV nkBT { ÅÇ k ÑÖ ÄÅ ÉÑ ÅÅ i Q y ÑÑ z Å j Ñ 1 − ϕB0ÅÅÅÅlogjjjj 0B zzzz + 1ÑÑÑÑ + ÅÅ j ϕB V z ÑÑ V { ÅÇ k ÑÖ

(31)

where PB1−BB is the number of ways of selecting NBB donors for self-association out of all donors in B* sticker groups, PB2−BB the number of ways of selecting NBB acceptors for selfassociation out of all acceptors in B* sticker groups, PBpair−BB the number of ways of pairing NBB B* donors with NBB B* acceptors, PB1−AB the number of ways of selecting NAB donors out of the remaining NAst−NBB donors in B* sticker groups for interassociation, PA−AB the number of ways of selecting NAB acceptors out of all acceptor groups in A* stickers for interassociation, and Ppair−AB the number of ways of pairing donors and acceptors for interassociation bonds. These quantities are PB1 − BB =

PB2 − BB =

NstB! NBB! (NstB −

PA − AB =

ϕ0A

NBB)!

(32)

∂qi(r, s) ∂s

(NstB − NBB)!

Ppair − AB = NAB!

(40)

= R G 2∇2 qi(r, s) − Wi (r)qi(r, s)

(41)

with the boundary condition qi(r, 0) = 1. Here, RG is the radius of gyration of the unperturbed chain. f int(r) is given by

(35)

fint (r) = χϕA(r)ϕB(r) + fst (r)

NstA! NAB! (NstA − NAB)!

∫ qi(r, 1) d3r

where qi(r,s) is proportional to the probability of finding the segment corresponding to the contour index s of the polymer i (i = A or B) at position r. qi(r,s) satisfies the equation

(33) (34)

NAB! (NstB − NBB − NAB)!

(39)

ϕ0B

where and are the the overall volume fractions of A and B segments in the system, respectively, and ξ(r) is a field that enforces the incompressibility condition that the sum of volume fractions, ϕA(r) and ϕB(r), is 1. QA and QB are the partition functions of ideal A and B chains in external fields WA(r) and WB(r) and are obtained from the relation Q i[wi ] =

PBpair − BB = NBB!

PB1 − AB =

∫ d3r[−WA(r)ϕA(r) − WB(r)ϕB(r) 1 − Ξ(r)(1 − ϕA(r) − ϕB(r))] + ∫ d3rNfint (r) V ×

NstB! NBB! (NstB − NBB)!

Article

(42)

Implicit in eq 39 is the assumption that the free energy of the system is a functional of the local free energy density and can be split into two parts: (a) the entropy of a system of noninteracting A and B homopolymers in external fields and (b) the free energy of short-range interactions in the local homogeneity approximation. We note that in a more rigorous

(36) (37)

Now, proceeding as we did in the previous section, we get L

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

National Institute of Standards and Technology, as part of the Center for Hierarchical Materials Design (CHiMaD).

SCFT formalism one needs to consider all possible architectures of chains resulting from association and include their partition functions (Q) in the free energy expression. Such approaches have been applied to simplistic cases such as end-functionalized polymer chains,42−45 where association results in diblock, triblock, and graft copolymers from homopolymer chains. Schmid77 developed an SCFT approach suitable for irreversibly cross-linked networks by introducing the cross-link positions as new degrees of freedom. As an application of this theory, the author studied the order− disorder transition in networks of AB diblock copolymers where the chain ends (A ends and B ends) form the cross-links, with A ends joining A blocks and B ends joining B blocks. In our system, the bonds are thermoreversible, meaning the system is able to access all conformations and is not frozen in any particular conformational state. If hydrogen bonds are randomly distributed, one could assume that in the thermodynamic limit the free energy of the quenched disorder is the same as that of an annealed disorder, as was shown in the case of fully random copolymers AB.57,58 In any event, it is difficult to adapt the SCFT theory of Schmid77 for fixed network topology to our case because of the difficulty in determining the partition functions of quenched configurations, whose number grows factorially with the number of hydrogen bonded groups per chain. In order to obtain the equilibrium density and chemical fields, one minimizes the functional F with respect to all fields. Setting the variational derivatives of F with respect to WA(r), WB(r), ϕA(r), and ϕB(r) to zero results in ϕA(r) =

ϕB(r) =

ϕA0V QA ϕB0V QB

∫0

■

(1) Walsh, D.; Rostami, S. Key Polymers Properties and Performance; Springer: 1985; pp 119−169. (2) Kuo, S. W.; Chang, F. C. Studies of miscibility behavior and hydrogen bonding in blends of poly (vinylphenol) and poly (vinylpyrrolidone). Macromolecules 2001, 34, 5224−5228. (3) Zhang, X.; Takegoshi, K.; Hikichi, K. Poly (vinylphenol)/poly (methyl acrylate) and poly (vinylphenol)/poly (methyl methacrylate) blends: Hydrogen bonding, miscibility, and blending effects on molecular motions as studied by carbon-13 CP/MAS NMR. Macromolecules 1991, 24, 5756−5762. (4) Meaurio, E.; Zuza, E.; Sarasua, J.-R. Miscibility and specific interactions in blends of poly (L-lactide) with poly (vinylphenol). Macromolecules 2005, 38, 1207−1215. (5) He, Y.; Zhu, B.; Inoue, Y. Hydrogen bonds in polymer blends. Prog. Polym. Sci. 2004, 29, 1021−1051. (6) Cao, X.; Jiang, M.; Yu, T. Controllable specific interaction and miscibility in polymer blends, 1. Hydrogen bonding and morphology. Makromol. Chem. 1989, 190, 117−128. (7) Huang, M.-W.; Kuo, S.-W.; Wu, H.-D.; Chang, F.-C.; Fang, S.-Y. Miscibility and hydrogen bonding in blends of poly (vinyl acetate) with phenolic resin. Polymer 2002, 43, 2479−2487. (8) Kuo, S. W.; Chang, F. C. Miscibility and hydrogen bonding in blends of poly (vinylphenol-co-methyl methacrylate) with poly (ethylene oxide). Macromolecules 2001, 34, 4089−4097. (9) Viswanathan, S.; Dadmun, M. D. Guidelines to creating a true molecular composite: inducing miscibility in blends by optimizing intermolecular hydrogen bonding. Macromolecules 2002, 35, 5049− 5060. (10) He, Y.; Zhu, B.; Inoue, Y. Hydrogen bonds in polymer blends. Prog. Polym. Sci. 2004, 29, 1021−1051. (11) Zhao, J.; Pearce, E.; Kwei, T.; Jeon, H.; Kesani, P.; Balsara, N. Micelles formed by a model hydrogen-bonding block copolymer. Macromolecules 1995, 28, 1972−1978. (12) Matsushita, Y. Creation of hierarchically ordered nanophase structures in block polymers having various competing interactions. Macromolecules 2007, 40, 771−776. (13) Chen, J.; Yan, B.; Wang, X.; Huang, Q.; Thundat, T.; Zeng, H. Core cross-linked double hydrophilic block copolymer micelles based on multiple hydrogen-bonding interactions. Polym. Chem. 2017, 8, 3066−3073. (14) Pryamitsyn, V.; Han, S. H.; Kim, J. K.; Ganesan, V. Curvature Modification of Block Copolymer Microdomains Using Blends of Block Copolymers with Hydrogen Bonding Interactions. Macromolecules 2012, 45, 8729−8742. (15) Chen, S.-C.; Kuo, S.-W.; Jeng, U.-S.; Su, C.-J.; Chang, F.-C. On modulating the phase behavior of block copolymer/homopolymer blends via hydrogen bonding. Macromolecules 2010, 43, 1083−1092. (16) Han, S. H.; Pryamitsyn, V.; Bae, D.; Kwak, J.; Ganesan, V.; Kim, J. K. Highly asymmetric lamellar nanopatterns via block copolymer blends capable of hydrogen bonding. ACS Nano 2012, 6, 7966−7972. (17) Han, S. H.; Kim, J. K.; Pryamitsyn, V.; Ganesan, V. Phase Behavior of Binary Blends of Block Copolymers Having Hydrogen Bonding. Macromolecules 2011, 44, 4970−4976. (18) Chen, Y.-T.; Lo, C.-T. Self-assembled structures in block copolymer/graft copolymer blends with hydrogen bonding interaction. Soft Matter 2013, 9, 1756−1760. (19) Salim, N. V.; Hameed, N.; Guo, Q. Competitive hydrogen bonding and self-assembly in poly (2-vinyl pyridine)-block-poly (methyl methacrylate)/poly (hydroxyether of bisphenol A) blends. J. Polym. Sci., Part B: Polym. Phys. 2009, 47, 1894−1905. (20) Dobrosielska, K.; Wakao, S.; Takano, A.; Matsushita, Y. Nanophase-separated structures of AB block copolymer/C homopolymer blends with complementary hydrogen-bonding interactions. Macromolecules 2008, 41, 7695−7698.

1

∫0

ds qA(r, s)qA(r, 1 − s)

(43)

1

ds qB(r, s)qB(r, 1 − s)

WA(r) = χNϕB(r) + N

WB(r) = χNϕA(r) + N

dfst dϕA(r) dfst dϕB(r)

Note that in the computation of

(44)

+ Ξ(r) (45)

+ Ξ(r) (46) dfst dϕA

ϕB is treated as being

independent of ϕA. The fields WA(r) and WB(r) and the density profiles are determined self-consistently. The method to do so is well established, and we do not discuss them here.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Monica Olvera de la Cruz: 0000-0002-9802-3627 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was performed under the financial assistance award DMR-1611076 from the National Science Foundation and 70NANB14H012 from the U.S. Department of Commerce, M

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

(42) Feng, E. H.; Lee, W. B.; Fredrickson, G. H. Supramolecular diblock copolymers: a field-theoretic model and mean-field solution. Macromolecules 2007, 40, 693−702. (43) Lee, W. B.; Elliott, R.; Katsov, K.; Fredrickson, G. H. Phase morphologies in reversibly bonding supramolecular triblock copolymer blends. Macromolecules 2007, 40, 8445−8454. (44) Zhang, X.; Wang, L.; Jiang, T.; Lin, J. Phase behaviors of supramolecular graft copolymers with reversible bonding. J. Chem. Phys. 2013, 139, 184901. (45) Mester, Z.; Lynd, N. A.; Delaney, K. T.; Fredrickson, G. H. Phase Coexistence Calculations of Reversibly Bonded Block Copolymers: A Unit Cell Gibbs Ensemble Approach. Macromolecules 2014, 47, 1865−1874. (46) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: 1953. (47) Stockmayer, W. H. Theory of molecular size distribution and gel formation in branched-chain polymers. J. Chem. Phys. 1943, 11, 45−55. (48) Stockmayer, W. H. Theory of molecular size distribution and gel formation in branched polymers II. General cross linking. J. Chem. Phys. 1944, 12, 125−131. (49) Bekiranov, S.; Bruinsma, R.; Pincus, P. Solution behavior of polyethylene oxide in water as a function of temperature and pressure. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 55, 577. (50) Erukhimovich, I.; Thamm, M. V.; Ermoshkin, A. V. Theory of the Sol- Gel Transition in Thermoreversible Gels with Due Regard for the Fundamental Role of Mesoscopic Cyclization Effects. 1. Thermodynamic and Structural Characteristics of the Gel Phase. Macromolecules 2001, 34, 5653−5674. (51) Mester, Z.; Mohan, A.; Fredrickson, G. H. Macro-and microphase separation in multifunctional supramolecular polymer networks. Macromolecules 2011, 44, 9411−9423. (52) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: 1979. (53) Erbas, A.; Horinek, D.; Netz, R. R. Viscous Friction of Hydrogen-Bonded Matter. J. Am. Chem. Soc. 2012, 134, 623−630. (54) Erbas, A.; Netz, R. R. Confinement-Dependent Friction in Peptide Bundles. Biophys. J. 2013, 104, 1285−1295. (55) Pryamitsyn, V. A.; Kwon, H.-K.; Zwanikken, J. W.; Olvera de La Cruz, M. Anomalous phase behavior of ionic polymer blends and ionic copolymers. Macromolecules 2017, 50, 5194−5207. (56) Helfand, E.; Tagami, Y. Theory of the interface between immiscible polymers. J. Polym. Sci., Part B: Polym. Lett. 1971, 9, 741− 746. (57) Fredrickson, G. H.; Milner, S. T. Thermodynamics of random copolymer melts. Phys. Rev. Lett. 1991, 67, 835. (58) Fredrickson, G. H.; Milner, S. T.; Leibler, L. Multicritical phenomena and microphase ordering in random block copolymers melts. Macromolecules 1992, 25, 6341−6354. (59) Swift, B.; De la Cruz, M. O. Random copolymers in concentrated solutions. EPL (Europhysics Letters) 1996, 35, 487. (60) Thelen, J. L.; Inceoglu, S.; Venkatesan, N. R.; Mackay, N. G.; Balsara, N. P. Relationship between ion dissociation, melt morphology, and electrochemical performance of lithium and magnesium single-ion conducting block copolymers. Macromolecules 2016, 49, 9139−9147. (61) Kumar, R.; Sumpter, B. G.; Muthukumar, M. Enhanced phase segregation induced by dipolar interactions in polymer blends. Macromolecules 2014, 47, 6491−6502. (62) Sing, C. E.; Zwanikken, J. W.; De La Cruz, M. O. Theory of melt polyelectrolyte blends and block copolymers: Phase behavior, surface tension, and microphase periodicity. J. Chem. Phys. 2015, 142, 034902. (63) Dormidontova, E. E.; ten Brinke, G. Association behavior of binary polymer mixtures under elongational flow. J. Chem. Phys. 2000, 113, 4814−4826. (64) Balsara, N. Physical Properties of Polymers Handbook; American Institute of Physics: New York, 1996; Chapter 19.

(21) Teo, L.-S.; Chen, C.-Y.; Kuo, J.-F. Fourier transform infrared spectroscopy study on effects of temperature on hydrogen bonding in amine-containing polyurethanes and poly (urethane- urea) s. Macromolecules 1997, 30, 1793−1799. (22) Liu, H.; Zhang, J.; Dai, W.; Zhao, Y. Synthesis and selfassembly of a dual-responsive monocleavable ABCD star quaterpolymer. Polym. Chem. 2017, 8, 6865−6878. (23) Tian, S.; Liu, G.; Wang, X.; Wu, T.; Yang, J.; Ye, X.; Zhang, G.; Hu, J.; Liu, S. pH-Regulated Reversible Transition Between Polyion Complexes (PIC) and Hydrogen-Bonding Complexes (HBC) with Tunable Aggregation-Induced Emission. ACS Appl. Mater. Interfaces 2016, 8, 3693−3702. (24) Li, J.; Ejima, H.; Yoshie, N. Seawater-assisted self-healing of catechol polymers via hydrogen bonding and coordination interactions. ACS Appl. Mater. Interfaces 2016, 8, 19047−19053. (25) Chen, Y.; Guan, Z. Multivalent hydrogen bonding block copolymers self-assemble into strong and tough self-healing materials. Chem. Commun. 2014, 50, 10868−10870. (26) Ruokolainen, J.; Mäkinen, R.; Torkkeli, M.; Mäkelä, T.; Serimaa, R.; ten Brinke, G.; Ikkala, O. Switching supramolecular polymeric materials with multiple length scales. Science 1998, 280, 557−560. (27) Aida, T.; Meijer, E.; Stupp, S. Functional supramolecular polymers. Science 2012, 335, 813−817. (28) Sijbesma, R. P.; Beijer, F. H.; Brunsveld, L.; Folmer, B. J.; Hirschberg, J. K.; Lange, R. F.; Lowe, J. K.; Meijer, E. Reversible polymers formed from self-complementary monomers using quadruple hydrogen bonding. Science 1997, 278, 1601−1604. (29) Zhou, J.; Shi, A.-C. Microphase separation induced by differential interactions in diblock copolymer/homopolymer blends. J. Chem. Phys. 2009, 130, 234904. (30) Tang, C.; Lennon, E. M.; Fredrickson, G. H.; Kramer, E. J.; Hawker, C. J. Evolution of block copolymer lithography to highly ordered square arrays. Science 2008, 322, 429−432. (31) Dehghan, A.; Shi, A.-C. Modeling hydrogen bonding in diblock copolymer/homopolymer blends. Macromolecules 2013, 46, 5796− 5805. (32) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: New York, 2003; Vol. 23. (33) Painter, P. C.; Park, Y.; Coleman, M. M. Hydrogen bonding in polymer blends. 2 Theory. Macromolecules 1988, 21, 66−72. (34) Mailhot, B.; Komvopoulos, K.; Ward, B.; Tian, Y.; Somorjai, G. Mechanical and friction properties of thermoplastic polyurethanes determined by scanning force microscopy. J. Appl. Phys. 2001, 89, 5712−5719. (35) Bourara, H.; Hadjout, S.; Benabdelghani, Z.; Etxeberria, A. Miscibility and hydrogen bonding in blends of poly (4-vinylphenol)/ poly (vinyl methyl ketone). Polymers 2014, 6, 2752−2763. (36) Semenov, A. N.; Rubinstein, M. Thermoreversible gelation in solutions of associative polymers. 1. Statics. Macromolecules 1998, 31, 1373−1385. (37) Dormidontova, E. E. Role of competitive PEO- water and water- water hydrogen bonding in aqueous solution PEO behavior. Macromolecules 2002, 35, 987−1001. (38) Parthasarathi, R.; Bellesia, G.; Chundawat, S.; Dale, B.; Langan, P.; Gnanakaran, S. Insights into hydrogen bonding and stacking interactions in cellulose. J. Phys. Chem. A 2011, 115, 14191−14202. (39) Sheridan, R. P.; Lee, R. H.; Peters, N.; Allen, L. C. Hydrogenbond cooperativity in protein secondary structure. Biopolymers 1979, 18, 2451−2458. (40) Qian, X. The effect of cooperativity on hydrogen bonding interactions in native cellulose Iβ from ab initio molecular dynamics simulations. Mol. Simul. 2008, 34, 183−191. (41) Otero, R.; Schöck, M.; Molina, L. M.; Lægsgaard, E.; Stensgaard, I.; Hammer, B.; Besenbacher, F. Guanine quartet networks stabilized by cooperative hydrogen bonds. Angew. Chem., Int. Ed. 2005, 44, 2270−2275. N

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (65) Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneko, M. Upper and lower critical solution temperatures in poly (ethylene glycol) solutions. Polymer 1976, 17, 685−689. (66) Coleman, M.; Moskala, E. FTi. r. studies of polymer blends containing the poly (hydroxy ether of bisphenol A) and poly (εcaprolactone). Polymer 1983, 24, 251−257. (67) McMaster, L. P. Copolymers, Polyblends, and Composites; 1975; Chapter 5, pp 43−65. (68) Matsuyama, A.; Tanaka, F. Theory of solvation-induced reentrant phase separation in polymer solutions. Phys. Rev. Lett. 1990, 65, 341. (69) Clarke, C. J.; Eisenberg, A.; LaScala, J.; Rafailovich, M. H.; Sokolov, J.; Li, Z.; Qu, S.; Nguyen, D.; Schwarz, S. A.; Strzhemechny, Y.; Sauer, B. B. Measurements of the Flory-Huggins interaction parameter for polystyrene-poly(4-vinylpyridine) blends. Macromolecules 1997, 30, 4184−4188. (70) Olvera de la Cruz, M.; Edwards, S.; Sanchez, I. Concentration fluctuations in polymer blend thermodynamics. J. Chem. Phys. 1988, 89, 1704−1708. (71) Wang, Z.-G. Concentration fluctuation in binary polymer blends: χ parameter, spinodal and Ginzburg criterion. J. Chem. Phys. 2002, 117, 481−500. (72) Grzywacz, P.; Qin, J.; Morse, D. C. Renormalization of the oneloop theory of fluctuations in polymer blends and diblock copolymer melts. Phys. Rev. E 2007, 76, 061802. (73) Fredrickson, G. H.; Helfand, E. Fluctuation effects in the theory of microphase separation in block copolymers. J. Chem. Phys. 1987, 87, 697−705. (74) Mayes, A.; De La Cruz, M. O. Concentration fluctuation effects on disorder-order transitions in block copolymer melts. J. Chem. Phys. 1991, 95, 4670−4677. (75) Liang, H.; Favis, B.; Yu, Y.; Eisenberg, A. Correlation between the interfacial tension and dispersed phase morphology in interfacially modified blends of LLDPE and PVC. Macromolecules 1999, 32, 1637−1642. (76) Matsen, M. W. Self-consistent field theory and its applications. Soft Matter 2005, 1, 87−178. (77) Schmid, F. Self-consistent field approach for cross-linked copolymer materials. Phys. Rev. Lett. 2013, 111, 028303.

O

DOI: 10.1021/acs.macromol.8b00661 Macromolecules XXXX, XXX, XXX−XXX